Finite Group Databasebeta

← #16931785

Subgroups — #16931785

Order 4352.

526 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531387C211yesmaximal, characteristic
2#14527304C212yesmaximal
3#14527114C211yesmaximal, characteristic
4#14526670C212yesmaximal
5C17×#2583C211yesmaximal, characteristic
6#11762773C2212yes—
7#11762695—24no—
8#11762672—21no—
9#11762669—21no—
10C4×C22⋊Dic17C2212yes—
11#11762546—24no—
12#11762535—22no—
13C2×C17×C22⋊Q8C2211yescharacteristic
14C22×C4⋊C4×C17C2211yescharacteristic
15C17×#398C412yescomplemented
16C17×#395C412yescomplemented
17C17×#389—22no—
18C17×#385—21no—
19C17×#385—21no—
20C4×C22⋊C4×C17C2211yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17C2311yescharacteristic
25C17×C22⋊Q8—22no—
26C17×C22⋊Q8—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17D412yes—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C42×C17—21no—
38C2×C22⋊Dic17—22no—
39C2×C4⋊Dic17—22no—
40C2×C4×Dic17—22no—
41C2×#63223—22no—
42C17×#95—22no—
43C17×#95—22no—
44C17×#95C2×C412yes—
45#10587502—22no—
46#10587502—24no—
47#10587502D414yes—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C24×C17C2×D411yescharacteristic
51C2×Q8×C17—22no—
52C2×Q8×C17—81no—
53C22×C4×C17—22no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—81no—
62C22×C4×C17—81no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D412yes—
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17#4112yes—
69C22×Dic17—24no—
70C4⋊C4×C17—24no—
71C4⋊C4×C17—24no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—44no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—82no—
76C22⋊C4×C17—24no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—41no—
79C42×C17—81no—
80C22⋊Dic17—44no—
81C4⋊Dic17—82no—
82C4×Dic17—82no—
83#63223—82no—
84C2×#3622—42no—
85#9412—171nomaximal, Sylow
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—41no—
89C23×C17C2×C22⋊C411yescharacteristic
90C23×C17#12011yescharacteristic
91C23×C17#12311yescharacteristic
92Q8×C17—82no—
93C2×C4×C17—24no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—81no—
113C2×C4×C17C4×D412yes—
114C2×C4×C17#12112yes—
115C2×Dic17—44no—
116C2×Dic17—44no—
117C2×Dic17—44no—
118#3622—82no—
119#2583D1711yescharacteristic
120#2313—171no—
121#1508—172no—
122#1463—171no—
123#1297—172no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—44no—
132C22×C17#38611yescharacteristic
133C22×C17#42112yes—
134C22×C17#45612yes—
135C4×C17—22no—
136C4×C17—22no—
137C4×C17—44no—
138C4×C17—42no—
139C4×C17—42no—
140C4×C17—81no—
141C4×C17—81no—
142Dic17—84no—
143C2×C22⋊Q8C2×D1711yescharacteristic
144C22×C4⋊C4C2×D1711yescharacteristic
145#407—172no—
146#398Dic1712yescomplemented
147#395Dic1712yescomplemented
148#389—22no—
149#387—344no—
150#385—21no—
151#385—21no—
152#381—341no—
153#380—341no—
154C4×C22⋊C4—172no—
155C4×C22⋊C4C2×D1711yescharacteristic
156#328—344no—
157#324—342no—
158C2×C17—22no—
159C2×C17—22no—
160C2×C17—41no—
161C2×C17#188611yescharacteristic
162C2×C17#189711yescharacteristic
163C2×C17#191311yescharacteristic
164C22×Q8C2×Dic1711yescharacteristic
165C23×C4C22⋊D1711yescharacteristic
166C23×C4C22⋊D1711yescharacteristic
167C23×C4C22×D1711yescharacteristic
168C2×M16—342no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—342no—
177C2×C4⋊C4C2×Dic1711yescharacteristic
178C2×C4⋊C4C2×Dic1712yes—
179C2×C4⋊C4C22⋊D1712yes—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—342no—
183C2×C42—21no—
184C2×C42—342no—
185C22⋊C8—342no—
186C22⋊C8—342no—
187#95—22no—
188#95—22no—
189#95—174no—
190#95—342no—
191#95—344no—
192#95C2×Dic1712yes—
193C17#941211yescharacteristic, complemented, Sylow
194C24C2×C22⋊D1711yescharacteristic
195C2×Q8—22no—
196C2×Q8—81no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—344no—
208C22×C4C22⋊Dic1712yes—
209C22×C4C22⋊Dic1712yes—
210C22×C4D4×D1712yes—
211C22×C4#6325412yes—
212C22×C4C22×Dic1711yescharacteristic
213C22×C4C2×C22⋊D1711yescharacteristic
214M16—1362no—
215C2×C8—682no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—1362no—
223C22⋊C4—24no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—684no—
227C42—81no—
228C42—1362no—
229C23—22no—
230C23—22no—
231C23—41no—
232C23#1058762111yescharacteristic
233C23#1058762311yescharacteristic
234C23C2×C22⋊Dic1711yescharacteristic
235Q8—82no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—684no—
257C2×C4—684no—
258C2×C4—684no—
259C2×C4D4×Dic1712yes—
260C2×C4#1058762512yes—
261C8—1362no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1176268911yescharacteristic
271C22#1176282612yes—
272C22#1176299112yes—
273C4—22no—
274C4—22no—
275C4—44no—
276C4—42no—
277C4—42no—
278C4—81no—
279C4—81no—
280C4—1364no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#1452917811yescharacteristic
285C2#1452923311yescharacteristic
286C2#1452932711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2583C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2583D17
hypocenterC17#9412
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. C17×#398C4semidirect
16. C17×#395C4semidirect
146. #398Dic17semidirect
147. #395Dic17semidirect
193. C17#9412semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145313872
#145273042
#145271142
#145266702
C17×#25832
#941217